Since the discovery of the fractional quantum Hall effect (FQHE) in 1982, topological phases of electrons have been a subject of great interest. Many abelian topological phases have been discovered in the context of the quantum Hall regime. More recently, high-temperature superconductivity and other complex materials have provided the impetus for further theoretical studies of and experimental searches for abelian topological phases. The types of microscopic models admitting such phases are now better understood. Much less is known about non-abelian topological phases. They are reputed to be obscure and complicated, and there has been little experimental motivation to consider non-abelian topological phases. However, non-abelian topological states would be an attractive milieu for quantum computation.
It has become increasingly clear that if a new generation of computers could be built to exploit quantum mechanical superpositions, enormous technological implications would follow. In particular, solid state physics, chemistry, and medicine would have a powerful new tool, and cryptography also would be revolutionized.
The standard approach to quantum computation is predicated on the quantum bit (“qubit”) model in which one anticipates computing on a local degree of freedom such as a nuclear spin. In a qubit computer, each bit of information is typically encoded in the state of a single particle, such as an electron or photon. This makes the information vulnerable. If a disturbance in the environment changes the state of the particle, the information is lost forever. This is known as decoherence—the loss of the quantum character of the state (i.e., the tendency of the system to become classical). All schemes for controlling decoherence must reach a very demanding and possibly unrealizable accuracy threshold to function.
Topology has been suggested to stabilize quantum information. A topological quantum computer would encode information not in the conventional zeros and ones, but in the configurations of different braids, which are similar to knots but consist of several different threads intertwined around each other. The computer would physically weave braids in space-time, and then nature would take over, carrying out complex calculations very quickly. By encoding information in braids instead of single particles, a topological quantum computer does not require the strenuous isolation of the qubit model and represents a new approach to the problem of decoherence.
In 1997, there were independent proposals by Kitaev and Freedman that quantum computing might be accomplished if the “physical Hilbert space” V of a sufficiently rich TQFT (topological quantum field theory) could be manufactured and manipulated. Hilbert space describes the degrees of freedom in a system. The mathematical construct V would need to be realized as a new and remarkable state for matter and then manipulated at will.
The computational power of a quantum mechanical Hilbert space is potentially far greater than that of any classical device. However, it is difficult to harness it because much of the quantum information contained in a system is encoded in phase relations which one might expect to be easily destroyed by its interactions with the outside world (“decoherence” or “error”). Therefore, error-correction is particularly important for quantum computation. Fortunately, it is possible to represent information redundantly so that errors can be diagnosed and corrected.
An interesting analogy with topology suggests itself: local geometry is a redundant way of encoding topology. Slightly denting or stretching a surface such as a torus does not change its genus, and small punctures can be easily repaired to keep the topology unchanged. Only large changes in the local geometry change the topology of the surface. Remarkably, there are states of matter for which this is more than just an analogy. A system with many microscopic degrees of freedom can have ground states whose degeneracy is determined by the topology of the system. The excitations of such a system have exotic braiding statistics, which is a topological effective interaction between them. Such a system is said to be in a topological phase. The unusual characteristics of quasiparticles in such states can lead to remarkable physical properties, such as a fractional quantized Hall conductance. Such states also have intrinsic fault-tolerance. Since the ground states are sensitive only to the topology of the system, interactions with the environment, which are presumably local, cannot cause transitions between ground states unless the environment supplies enough energy to create excitations which can migrate across the system and affect its topology. When the temperature is low compared to the energy gap of the system, such events will be exponentially rare.
A different problem now arises: if the quantum information is so well-protected from the outside world, then how can we—presumably part of the outside world—manipulate it to perform a computation? The answer is that we must manipulate the topology of the system. In this regard, an important distinction must be made between different types of topological phases. In the case of those states which are Abelian, we can only alter the phase of the state by braiding quasiparticles. In the non-Abelian case, however, there will be a set of g>1 degenerate states, ψa, a=1, 2, . . . , g of particles at x1, x2, . . . , xn. Exchanging particles 1 and 2 might do more than just change the phase of the wave function. It might rotate it into a different one in the space spanned by the ψaS:ψa→Mab12ψb  (1)
On the other hand, exchanging particles 2 and 3 leads to ψa→Mab23ψb. If Mab12 and Mab23 do not commute (for at least some pairs of particles), then the particles obey non-Abelian braiding statistics. In the case of a large class of states, the repeated application of raiding transformations Mabij allows one to approximate any desired unitary transformation to arbitrary accuracy and, in this sense, they are universal quantum computers. Unfortunately, no non-Abelian topological states have been unambiguously identified so far. Some proposals have been put forward for how such states might arise in highly frustrated magnets, where such states might be stabilized by very large energy gaps on the order of magnetic exchange couplings, but the best prospects in the short run are in quantum Hall systems, where Abelian topological phases are already known to exist.
The best candidate is the quantized Hall plateau with
      σ    ab    =            5      2        ⁢                            e          2                h            .      The 5/2 fractional quantum Hall state (as well as its particle-hole symmetric analog, the 7/2 state) is now routinely observed in high-quality (i.e., low-disorder) samples. In addition, extensive numerical work using finite-size diagonalization and wavefunction overlap calculations indicates that the 5/2 state belongs to the non-Abelian topological phase characterized by a Pfaffian quantum Hall wavefunction. The set of transformations generated by braiding quasiparticle excitations in the Pfaffian state is not computationally universal (i.e., is not dense in the unitary group), but other non-Abelian states in the same family are. Thus, it is important to (a) determine if the v= 5/2 state is, indeed, in the Pfaffian universality class and, if so, to (b) use it to store and manipulate quantum information.